cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A338327 a(n) is the least number k such that there are exactly n biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.

Original entry on oeis.org

1, 2, 14, 36, 234, 3510, 211297, 487425, 20136429
Offset: 0

Views

Author

Amiram Eldar, Oct 22 2020

Keywords

Comments

a(n) is the least k such that A338326(k) = n.
Dehkordi (1998) proved that for each k>=0 the sequence of numbers m such that A338326(m) = k has a positive asymptotic density. Therefore, this sequence is infinite.
a(9) > 10^10. - Bert Dobbelaere, Oct 29 2020

Examples

			a(0) = 1 since there are no biquadratefree powerful numbers between 1^2 = 1 and 2^2 = 4.
a(1) = 2 since there is one biquadratefree powerful number, 8 = 2^3, between 2^2 = 4 and 3^2 = 8.
a(2) = 14 since there are 2 biquadratefree powerful numbers, 200 = 2^3 * 5^2 and 216 = 2^3 * 3^3, between 14^2 = 196 and 15^2 = 225.

Links

Crossrefs

Cf. A119242, A337737, A338325, A338326.

Programs

  • Mathematica
    bqfpowQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], MemberQ[{2, 3 }, #] &]; f[n_] := Count[Range[n^2 + 1, (n + 1)^2 - 1], _?bqfpowQ]; mx = 5; s = Table[0, {mx}]; c = 0; n = 1; While[c < mx, i = f[n] + 1; If[i <= mx && s[[i]] == 0, c++; s[[i]] = n]; n++]; s

Extensions

a(8) from Bert Dobbelaere, Oct 29 2020

A338387 Numbers k such that there are no biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 12, 13, 15, 16, 17, 20, 21, 23, 24, 26, 27, 28, 29, 30, 32, 34, 35, 38, 39, 40, 41, 43, 44, 45, 47, 49, 50, 54, 56, 60, 61, 62, 63, 64, 66, 68, 69, 71, 74, 75, 76, 77, 79, 80, 81, 83, 84, 85, 86, 89, 90, 91, 95, 97, 99, 100, 101, 105, 106, 107
Offset: 1

Views

Author

Amiram Eldar, Oct 23 2020

Keywords

Comments

Positions of 0's in A338326.
The asymptotic density of this sequence is 0.623265038... (Dehkordi, 1998).

Examples

			1 is a term since the two numbers between 1^2 = 1 and (1+1)^2 = 4, 2 and 3, are not biquadratefree powerful.
2 is not a term since there is a biquadratefree powerful number, 8 = 2^3, between 2^2 = 4 and (2+1)^2 = 9.

Links

Crossrefs

Cf. A336175, A338325, A338326, A338327, A338388, A338389, A338390, A338391, A338392.

Programs

  • Mathematica
    bqfpowQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], MemberQ[{2, 3}, #] &]; Select[Range[100], !AnyTrue[Range[#^2 + 1, (# + 1)^2 - 1], bqfpowQ] &]

A338388 Numbers k such that there is a single biquadratefree powerful number (A338325) between k^2 and (k+1)^2.

Original entry on oeis.org

2, 5, 8, 10, 11, 18, 19, 22, 25, 33, 37, 42, 46, 48, 51, 52, 53, 55, 57, 58, 59, 65, 70, 73, 78, 87, 88, 92, 94, 96, 102, 103, 104, 109, 111, 114, 115, 116, 119, 121, 122, 135, 144, 145, 149, 150, 155, 157, 164, 165, 166, 176, 181, 182, 183, 185, 190, 191, 195
Offset: 1

Views

Author

Amiram Eldar, Oct 23 2020

Keywords

Comments

Positions of 1's in A338326.
The asymptotic density of this sequence is 0.308276695... (Dehkordi, 1998).

Examples

			2 is a term since there is a single biquadratefree powerful number, 8 = 2^3, between 2^2 = 4 and (2+1)^2 = 9.

Links

Crossrefs

Cf. A336176, A338325, A338326, A338327, A338387, A338389, A338390, A338391, A338392.

Programs

  • Mathematica
    bqfpowQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], MemberQ[{2, 3}, #] &]; Select[Range[200], Count[Range[#^2 + 1, (# + 1)^2 - 1], _?bqfpowQ] == 1 &]

A338389 Numbers k such that there are exactly two biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.

Original entry on oeis.org

14, 31, 67, 72, 82, 93, 98, 110, 132, 140, 156, 172, 189, 192, 223, 240, 257, 281, 285, 322, 347, 368, 379, 407, 410, 414, 426, 441, 455, 468, 472, 481, 488, 514, 515, 517, 524, 525, 537, 551, 555, 574, 579, 602, 613, 664, 680, 693, 702, 703, 737, 743, 749, 755
Offset: 1

Views

Author

Amiram Eldar, Oct 23 2020

Keywords

Comments

Positions of 2's in A338326.
The asymptotic density of this sequence is 0.058757863... (Dehkordi, 1998).

Examples

			14 is a term since there are exactly two biquadratefree powerful numbers, 200 = 2*3 * 5^2 and 216 = 2^3 * 3^3, between 14^2 = 196 and (14+1)^2 = 225.

Links

Crossrefs

Cf. A336177, A338325, A338326, A338327, A338387, A338388, A338390, A338391, A338392.

Programs

  • Mathematica
    bqfpowQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], MemberQ[{2, 3}, #] &]; Select[Range[800], Count[Range[#^2 + 1, (# + 1)^2 - 1], _?bqfpowQ] == 2 &]

A338390 Numbers k such that there are exactly three biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.

Original entry on oeis.org

36, 161, 364, 659, 722, 771, 896, 911, 981, 987, 1241, 1359, 1486, 1575, 1822, 2042, 2090, 2435, 2537, 2582, 2733, 2870, 2873, 2967, 2983, 3012, 3101, 3108, 3198, 3222, 3278, 3419, 3465, 3544, 3668, 3855, 3860, 3934, 4024, 4092, 4188, 4426, 4437, 4494, 4511, 4522
Offset: 1

Views

Author

Amiram Eldar, Oct 23 2020

Keywords

Comments

Positions of 3's in A338326.
The asymptotic density of this sequence is 0.008234579... (Dehkordi, 1998).

Examples

			36 is a term since there are exactly three biquadratefree powerful numbers, 1323 = 3^3 * 7^2, 1331 = 11^3 and 1352 = 2^3 * 13^2, between 36^2 = 1296 and (36+1)^2 = 1369.

Links

Crossrefs

Cf. A336178, A338325, A338326, A338327, A338387, A338388, A338389, A338391, A338392.

Programs

  • Mathematica
    bqfpowQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], MemberQ[{2, 3}, #] &]; Select[Range[1000], Count[Range[#^2 + 1, (# + 1)^2 - 1], _?bqfpowQ] == 3 &]

A338391 Numbers k such that there are exactly four biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.

Original entry on oeis.org

234, 675, 2426, 8075, 8391, 9093, 9548, 10214, 10340, 11213, 13816, 14523, 14970, 15593, 17329, 17803, 20649, 22483, 23020, 23128, 24842, 25971, 26318, 26557, 28241, 28677, 29124, 29837, 31058, 31338, 31732, 31907, 32490, 35676, 35765, 36302, 37599, 41077, 42577
Offset: 1

Views

Author

Amiram Eldar, Oct 23 2020

Keywords

Comments

Positions of 4's in A338326.
The asymptotic density of this sequence is 0.000089634... (Dehkordi, 1998).

Examples

			234 is a term since there are exactly four biquadratefree powerful numbers, 54872 = 2^3 * 19^3, 54925 = 5^2 * 13^3, 55112 = 2^3 * 83^2 and 55125 = 3^2 * 5^3 * 7^2, between 234^2 = 54756 and (234+1)^2 = 55225.

Links

Crossrefs

Cf. A338325, A338326, A338327, A338387, A338388, A338389, A338390, A338392.

Programs

  • Mathematica
    bqfpowQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], MemberQ[{2, 3}, #] &]; Select[Range[10^4], Count[Range[#^2 + 1, (# + 1)^2 - 1], _?bqfpowQ] == 4 &]

A338392 Numbers k such that there are exactly five biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.

Original entry on oeis.org

3510, 3611, 16871, 25076, 26910, 35810, 50501, 83107, 101287, 104686, 111836, 152924, 153433, 217983, 239163, 247301, 262413, 266282, 277635, 294453, 298950, 340228, 344510, 362830, 369877, 385336, 475063, 524827, 536793, 537713, 539293, 567062, 568609, 614283
Offset: 1

Views

Author

Amiram Eldar, Oct 23 2020

Keywords

Comments

Positions of 5's in A338326.
The asymptotic density of this sequence is 0.000011113... (Dehkordi, 1998).

Examples

			3510 is a term since there are exactly five biquadratefree powerful numbers, 12320648 = 2^3 * 17^2 * 73^2, 12321000 = 2^3 * 3^2 * 5^3 * 37^2, 12324500 = 2^2 * 5^3 * 157^2, 12325975 = 5^2 * 79^3 and 12326391 = 3^3 * 7^3 * 11^3, between 3510^2 = 12320100 and (3510+1)^2 = 12327121.

Links

Crossrefs

Cf. A338325, A338326, A338327, A338387, A338388, A338389, A338390, A338391.

Programs

  • Mathematica
    bqfpowQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], MemberQ[{2, 3}, #] &]; Select[Range[25000], Count[Range[#^2 + 1, (# + 1)^2 - 1], _?bqfpowQ] == 5 &]
Showing 1-7 of 7 results.

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